This paper classifies Ricci soliton subgroups in a specific type of nilpotent group.
problem Classifying Ricci soliton subgroups in a specific type of nilpotent group.
method Using the properties of nilpotent Iwasawa groups and Lie subgroups.
result Classification of codimension one Lie subgroups of nilpotent Iwasawa groups that are Ricci solitons.
This thesis was inspired by work of M. Cowling, F. De Mari, A. Koranyi and M. Reimann, who studied multicontact structures for the homogeneous manifolds G/P, where G is a semisimple Lie group and P is the minimal parabolic subgroup of G. The multicontact structure here arises naturally by the nilpotent component N of t…
The paper constructs pseudo-Iwasawa solvmanifolds with Killing spinors.
problem Constructing Killing spinors on pseudo-Riemannian solvmanifolds.
method Using nilsolitons and pseudo-Iwasawa condition, the paper constructs families of pseudo-Iwasawa solvmanifolds with Killing spinors.
result All pseudo-Iwasawa solvmanifolds admitting a Killing spinor belong to a specific family.
The paper studies automorphisms of 2-step nilpotent Lie groups, showing continuity up to center and field automorphisms.
problem Investigating the continuity of abstract automorphisms in 2-step nilpotent Lie groups.
method Analyzes various types of 2-step nilpotent Lie groups, using tools from Riemannian geometry.
result Abstract automorphisms are continuous 'up to discontinuity due to the center and field automorphisms of C' for many 2-step nilpotent Lie groups. We prove the finiteness of the cohomology of torsion-free lattices in a semisimple Lie group of real rank one with coefficients in the distribution vector globalization of Harish-Chandra modules. The cohomology is expressed in terms of automorphic and cusp forms. We also consider the Lie-algebra cohomology of these glo…
If f is a conformal mapping defined on a connected open subset of a Carnot group G, then either f is the composition of a translation, a dilation and an isometry, or G is the nilpotent Iwasawa component of a real rank 1 simple Lie group S, and f arises from the action of S on G, viewed as an open subset of S/P, where P…
Study models of Gödel Universe using Lie groups and Iwasawa decomposition.
problem Modeling the Gödel Universe as a Lie group with specific metrics.
method Iwasawa decomposition for semisimple Lie groups, left-invariant Lorentz metric on SL(2,R).
result Isometry between sub-Riemannian Lie groups induced by Iwasawa decomposition.
Classifies Ricci soliton subgroups in specific Lie groups.
problem Classifying Ricci soliton subgroups in solvable Lie groups.
method Analyzing codimension one subgroups of solvable Iwasawa groups and related spaces.
result Classifications of Ricci soliton subgroups in various Lie groups.
We set up an abstract framework that allows the investigation of Iwasawa decompositions for involutive infinite-dimensional Lie groups modeled on Banach spaces. As an application, we construct Iwasawa decompositions for classical real or complex Banach-Lie groups associated with the Schatten ideals ${\mathfrak S}_p({\m…
Constructs explicit p-harmonic functions on specific Lie groups.
problem Finding explicit p-harmonic functions on a specific class of Lie groups.
method Constructs explicit p-harmonic functions on rank-one Lie groups of Iwasawa type.
result Proves existence of proper p-harmonic functions on these groups.
Iwasawa manifold is a compact complex homogeneous manifold isomorphic to a quotient of the group of complex unipotent 3×3 matrices by a cocompact lattice. We prove that any compact complex curve in an Iwasawa manifold is contained in a holomorphic torus. We also prove that any Kähler surface in an Iwasawa mani…
Let J1 be the real form of a complex simple Jordan algebra such that the automorphism group is F4(−20). By using some orbit types of F4(−20) on J1, for F4(−20), explicitly, we give the Iwasawa decomposition, the Oshima--Sekiguchi's Kε−Iwasawa decomp…
Study on special metrics and deformations of solvmanifolds.
problem Existence and properties of Kähler metrics on solvmanifolds.
method Investigation of strong Kähler with torsion metrics and balanced metrics on deformations of specific solvmanifolds.
result Non-existence of certain metrics on specific solvmanifolds.
The paper constructs Einstein Sasaki metrics on solvable Lie groups.
problem Constructing left-invariant Einstein pseudo-Riemannian Sasaki metrics on solvable Lie groups.
method Characterizing pseudo-Kähler structures and derivations giving rise to Sasaki-Einstein metrics.
result Classification of z-standard Sasaki solvable Lie algebras of dimension ≤7. Study of knot invariants using twisted Iwasawa theory.
problem Determining knot properties like genus and fiberedness.
method Introducing twisted Iwasawa invariants for knot representations and covers.
result Iwasawa invariants determine knot properties and their rigidity.
The aim of this paper is to study cohomogeneity one isometric linear actions on the p+q-dimensional pseudo-Euclidean space Rp,q. It is proved that the natural isometric action of the nilpotent factor of an Iwasawa decomposition of SO(p,q) is not of cohomogeneity one. The orbits of cohomogeneity one ac…
Classifies CR submanifolds in complex hyperbolic spaces.
problem Understanding CR submanifolds in complex hyperbolic spaces.
method Classifying orbits of a subgroup of the solvable part of the Iwasawa decomposition.
result Classification of homogeneous CR submanifolds.
Analogues of Iwasawa invariants in the context of 3-dimensional topology have been studied by M.~Morishita and others. In this paper, following the dictionary of arithmetic topology, we formulate an analogue of Kida's formula on λ-invariants in a p-extension of Zp-fields for 3-manifolds. The proof is gi…
We derive expressions for the Ricci curvature tensor and scalar in terms of intrinsic torsion classes of half-flat manifolds by exploiting the relationship between half-flat manifolds and non-compact G2 holonomy manifolds. Our expressions are tested for Iwasawa and more general nilpotent manifolds. We also derive ex…
Geometrically proves Lie algebras are identified by their Iwasawa subalgebras.
problem Determining semi-simple Lie algebras from their subalgebras.
method Geometry of Einstein solvmanifolds and algebraic procedure.
result Semi-simple Lie algebras are uniquely determined by their Iwasawa subalgebras.
Study knot invariants using automorphism groups of free nilpotent groups.
problem Developing knot invariants using automorphism groups.
method Nilpotently p-localization of knot groups and automorphism groups of free nilpotent groups. result Maps from outer automorphism groups yield knot invariants.
The paper classifies and describes translators in SL(2,R) under specific symmetry conditions.
problem Classifying translators in SL(2,R) under invariant symmetry groups. method Analyzing translators invariant by one-parameter groups of isometries, using Iwasawa decomposition and Killing vector fields.
result Explicit parametrizations of translators are obtained for some cases.
Study of double complexes on Iwasawa manifold yields 3 isomorphism types.
problem Understanding the structure of double complexes on the Iwasawa manifold.
method Used Stelzig and Qi-Khovanov's structure theorem for double complexes.
result Identified and described exactly 3 isomorphism types of double complexes.
Study proves equality of LS-category and cohomological dimension for specific group homomorphisms.
problem Proving equality of LS-category and cohomological dimension for specific group homomorphisms.
method Analyzing epimorphisms and homomorphisms between specific types of almost nilpotent and virtually nilpotent groups.
result Equality of LS-category and cohomological dimension for specified group homomorphisms.
We compute the characteristic varieties and the Alexander polynomial of a finitely generated nilpotent group. We show that the first characteristic variety may be used to detect nilpotence. We use the Alexander polynomial to deduce that the only torsion-free, finitely generated nilpotent groups with positive deficiency…
Nilpotent quandles have simple characterizations and are closely related to nilpotency.
problem Characterizing and understanding nilpotent quandles.
method Characterization of generating sets, Hopf property, construction of free nilpotent quandles, simple presentations of associated groups.
result Nilpotent quandles have the Hopf property and are equivalent to reduced peripheral systems.
Based on the analogy between knots and primes, J. Hillman, D. Matei and M. Morishita defined the Iwasawa invariants for sequences of cyclic covers of links with an analogue of Iwasawa's class number formula of number fields. In this paper, we consider the existence of covers of links with prescribed Iwasawa invariants,…
Completes classification of G2-structures on specific nilpotent Lie groups.
problem Classifying seven-dimensional nilpotent Lie groups with purely coclosed G2-structures.
method Analyzing nilpotent Lie groups of various steps and dimensions.
result Classification of indecomposable 5- and 6-step nilpotent Lie groups.
Researchers determine Dehn functions of specific nilpotent groups.
problem Understanding the Dehn functions of central products of nilpotent groups.
method Analyzing families of filiform and Lie groups to determine Dehn functions.
result Confirms conjecture and provides evidence for lower Dehn functions in central products.
The paper studies gradings on nilpotent Lie algebras linked to smooth algebraic varieties.
problem Understanding gradings on nilpotent Lie algebras associated with algebraic varieties.
method Analyzing lattice structures in nilpotent Lie groups and their fundamental groups.
result Conditions for a lattice to be the fundamental group of a smooth complex algebraic variety.
This paper completes the classification of certain nilpotent Lie groups with specific geometric structures.
problem Classifying nilpotent Lie groups with purely coclosed G2-structures.
method Analyzing seven-dimensional nilpotent Lie groups of various steps.
result Classification of indecomposable 5- and 6-step nilpotent Lie groups with these structures.
Study torsion-free nilpotent fundamental groups of smooth varieties up to rank 7.
problem Characterize fundamental groups of smooth quasi-projective varieties.
method Analyzes fundamental groups of smooth quasi-projective varieties using topological and Lie group theory.
result Determine fundamental groups for smooth quasi-projective varieties up to rank 7.
We present explicit universal strict deformation quantization formulae for actions of Iwasawa subgroups AN of SU(1,n). This answers a question raised by Rieffel.
We study the question of the existence of left-invariant Sasaki contact structures on the seven-dimensional nilpotent Lie groups. It is shown that the only Lie group allowing Sasaki structure with a positive definite metric tensor is the Heisenberg group. We find a complete list of the 22 classes of seven-dimensional n…
Nilpotent Sasakian manifolds are Heisenberg nilmanifolds.
problem Characterizing Sasakian manifolds with nilpotent fundamental groups.
method Proved diffeomorphism to Heisenberg nilmanifolds.
result Compact aspherical Sasakian manifolds with nilpotent fundamental groups are Heisenberg nilmanifolds.
Gromov proposed an averaged version of the Dehn function and claimed that in many cases it should be subasymptotic to the Dehn function. Using results on random walks in nilpotent groups, we confirm this claim for most nilpotent groups. In particular, if a nilpotent group satisfies the isoperimetric inequality $δ(l)<Cl…
Study embeddings of 3-manifolds in S4 with nilpotent fundamental groups.
problem Embeddings of 3-manifolds in S4 with specific properties of fundamental groups. method Analyzes embeddings with nilpotent fundamental groups, determines groups with Hirsch length ≤5.
result Identifies all nilpotent groups with Hirsch length ≤5 and torsion-free.
A nonpolycyclic nilpotent-by-cyclic group Gamma can be expressed as the HNN extension of a finitely-generated nilpotent group N. The first main result is that quasi-isometric nilpotent-by-cyclic groups are HNN extensions of quasi-isometric nilpotent groups. The nonsurjective injection defining such an extension induces…
The paper establishes analogs of Stallings' theorem for group homomorphisms and their nilpotent quotients.
problem Understanding the structure of fundamental groups of geometric objects.
method Develops analogs of Stallings' theorem for group homomorphisms and their nilpotent quotients.
result Derives applications including non-isomorphic number fields and hyperbolic manifolds with isomorphic universal nilpotent quotients.
Criterion for nilpotent Lie groups to have nilsolitons.
problem Existence of nilsolitons in nilpotent Lie groups.
method Algebraic criterion for nilpotent Lie algebras, proving necessary and sufficient condition for nilsolitons.
result Criterion provides a necessary and sufficient condition for nilpotent Lie groups to admit nilsolitons.
Study limits of adjoint orbits for Lie groups, describing nilpotent orbits.
problem Understanding limits of adjoint orbits for Lie groups.
method Systematic and topological study of limits of continuous families of adjoint orbits for non-compact simple Lie groups.
result Explicit description of nilpotent orbits in terms of Richardson orbits for hyperbolic semisimple elements.
We identify the space of left-invariant oriented complex structures on the complex Heisenberg group, and prove that it has the homotopy type of the disjoint union of a point and a 2-sphere.
New representation for braid groups and surface braid groups, extending Lawrence-Krammer-Bigelow.
problem Constructing representations for braid groups and surface braid groups.
method Pro-nilpotent tower of representations, starting with the original LKB representation.
result 3-variable enrichment of the Lawrence-Krammer-Bigelow representation.
Global harmonic maps into SU(1,1) constructed from Smyth potentials using DPW method.
problem Globality of harmonic maps constructed from Smyth potentials in SU(1,1).
method Construct harmonic maps into SU(1,1) using the DPW method, solving a Riemann-Hilbert problem to achieve global Iwasawa factorization.
result Globality of the constructed harmonic maps proved using Bessel functions and asymptotic expansions.
New definition of Rumin complex for nilpotent Lie groups.
problem No new problem introduced.
method Alternative definition of Rumin complex on nilpotent Lie groups.
result Direct application of ℓq,p cohomology results to all nilpotent Lie groups. Study para-complex structures on specific Lie groups, finding explicit forms and properties.
problem Characterizing para-Kähler structures on six-dimensional nilpotent Lie groups.
method Examined left-invariant para-complex structures on six-dimensional nilpotent Lie groups, obtained explicit expressions and investigated curvature properties.
result Para-complex structures are nilpotent and para-Kähler metrics are Ricci-flat.
Study conformal Killing forms on specific nilpotent Lie groups.
problem Characterize conformal Killing forms on 2-step nilpotent Lie groups.
method Analyzing left-invariant forms on simply connected groups, proving properties of forms based on center dimension.
result Only specific forms exist under certain conditions.
The study constructs families of nilpotent Lie groups with geodesic orbit metrics.
problem Finding geodesic orbit metrics on nilpotent Lie groups.
method Construction of continuous families of nilpotent Lie groups.
result Continuous families of non-isomorphic nilpotent Lie groups with geodesic orbit metrics.